Dear Jean, In one of your messages, to Marta but with copies to many of us, you say: "And of course you may have found in some work of yours I do not know examples of Street fibrations which are neither equivalences nor Grothendieck fibrations. I'd greatly appreciate if you could give me such examples" We write now at this late stage to list some examples. For the purposes of this message, we shall call "abstract fibration" what some have been calling "Street fibration". (I) As defined in [J1], given a functor P : C --> X, an object c of a category C is said to be admissible if the induced functor C/c --> X/ Pc has a fully faithful right adjoint. In 1984 George did not think of Grothendieck fibrations and/or of abstract fibrations, and they are not mentioned in the paper, but to say that every object in C is admissible is exactly to say that P : C --> X is a abstract fibration. The main (in fact the unique) example considered in [J1] had: C = the opposite category of commutative rings (with 1); X = the category of Stone spaces; P = the Boolean spectrum functor, that is, Pc is the Stone space of the Boolean algebra of idempotents in c. This is a non-Grothendieck abstract fibration, and the results of [J1] applied to it gave Magid's Galois theory [M] as a special case (more details are given in [BJ]). (II) After a discussion with Saunders Mac Lane, George understood that in presenting classical Galois theory as a special case of his "categorical Galois Theory" it is often better to avoid the case of general commutative rings, and this led to the following example: C = the opposite category of finite-dimensional (as vector spaces) commutative k-algebras (with 1), where k is a fixed field; X = the category of finite sets; P = the Boolean spectrum functor, which, in this simple case, can be defined as Pc = the set of minimal (non-zero) idempotents in c. Again this is a non-Grothendieck abstract fibration. (III) The next most classical Galois theory is the one of covering spaces, and in order to get it George considers the following non- Grothendieck abstract fibration: C = E'tale(s), the category of E'tale spaces over a fixed locally connected space s (it is equivalent to the topos Shv(s)); X = the category of sets; P sends a bundle p: a --> s to the set of connected component of a. (By the way, replacing E'tale(s) with the category of all locally connected spaces with e'tale maps as morphisms would give an example of a abstract fibration which, unlike the previous examples, does not have a right adjoint.) (IV) It is easy to "put together" the examples considered in (II) and (III): Call an object c in an arbitrary category C with coproducts connected if the representable functor C(c,-) : C --> Sets preserves coproducts; then require C to be locally connected in the sense that every object in C is a coproduct of connected objects, and take: X = the category of sets; Pc = the set of "connected components" of c in the obvious sense. However, one should bear in mind the following: (i) the above construction as stated applies to (III), while for (II) one needs its finite version; and (ii) defining P as above requires certain choice. The choice problem disappears if C is defined as the category of families of objects of a certain category, but then P, defined as Pc = the index set of c, becomes a Grothendieck fibration. In concrete cases like (II) and (III) we have neither the choice problem, nor a Grothendieck fibration. There are a lot of such concrete cases: apart from many geometric/ topological examples, C can be, for example, the category of small categories, or the category of simplicial sets. Then again, if M is a monoid, C could be the category Set^M of sets equipped with an action of M; then the functor pi_0 sending an M-set to its set of connected components is a abstract fibration which is not a Grothendieck fibration. To summarize, there are many concrete categories equivalent to suitable categories of families that form a non-Grothendieck fibration over the categories of sets and of finite sets. Moreover, the concept of abstract fibration seems to be a good candidate for a general concept of connectedness and (which is almost the same) for a general concept of torsion theory, of which there are many concrete mathematical examples. (V) In particular, many of the examples of connectedness can use the category of Stone spaces as X (similarly to the commutative-ring- example in (I) above). The "algebraic" ones are studied in [D] (from a different viewpoint though), and a general approach to "Boolean admissibility" is developed in [CJ]. The simplest topological example is the reflection of compact Hausdorff spaces into Stone spaces, which again is a non-Grothendieck abstract fibration. (VI) There are localizations (finite limit preserving reflections) of course. Standard examples, as from abelian categories and toposes are "non-Grothendieck". (VII) Apart from Ross' reason for needing abstract fibration and George's use of "admissibility in categorical Galois theory", there is the "factorization system approach" developed in [CHK]. Fibrations are not mentioned there, but a reflection to a full subcategory is a abstract fibration if and only if it is semi-left-exact in the sense of [CHK]. And again, the examples mentioned in [CHK] are usually non- Grothendieck. The connection between "admissibility" and "semi-left- exactness" is explained in [CJKP] and in [JK]. The connection with fibrations is not explained anywhere, which is definitely an oversight. At one time Steve mentioned it to George, who said only something like "Maybe you will write about it somewhere ...." (VIII) In his further study of categorical Galois theory, George realized that a weaker notion of admissibility is more appropriate for his purposes. In the language of the theory of fibrations it would be formulated as "(not all, but only) morphisms from a given class have cartesian liftings" - and there are a lot of examples of Galois theories where this is useful. Those given classes might consist, for example, of regular epimorphisms in exact categories, E'tale maps of topological spaces, Kan fibrations of simplicial sets, etc. (IX) Some authors (e.g. Michel Thiebaud) studied functors P : C --> X, such that the induced functor C/c --> X/Pc is an equivalence for each c in C. There were no reasons to require such P to be Grothendieck fibrations (just like there is no reason to require localizations to be Grothendieck fibrations). (X) What we said above is surely incomplete. We might note that not all of the following points seem to be made in the literature: (a) Grothendieck fibrations, abstract fibrations, semi-left-exact reflections, and admissibility in categorical Galois theory are closely related to each other. (b) A reflective factorization system in the sense of [CHK] is essentially the same as a vertical-cartesian factorization system in the theory of fibrations. (c) Trivial coverings in categorical Galois theory are essentially the same as cartesian morphisms in the theory of fibrations. Thank you for your question. We hope our response is helpful. References [BJ] F. Borceux and G. Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, Cambridge, 2001 [CJ] A. Carboni and G. Janelidze, Boolean Galois theories, Georgian Math. J. 9, 4, 2002, 645-658 [CJKP] A. Carboni, G. Janelidze, G. M. Kelly, and R. Pare, On localization and stabilization of factorization systems, Applied Categorical Structures 5, 1997, 1-58 [CHK] C. Cassidy, M. H=E9bert, and G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. Ser. A 38 (1985) 287-329 [D] Y. Diers, Categories of Boolean sheaves of simple algebras, Lecture Notes in Mathematics 1187, Springer-Verlag, Berlin, 1986 [J1] G. Janelidze, Magid's theorem in categories, Bull. Georgian Acad. Sci. 114, 3, 1984, 497-500 [JK] G. Janelidze and G. M. Kelly, The reflectiveness of covering morphisms in algebra and geometry, Theory and Applications of Categories 3, 1997, 132-159 [M] A. R. Magid, The separable Galois theory of commutative rings, Marcel Dekker Inc., New York, 1974 With best regards, George Janelidze, Steve Lack and Ross Street. 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